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[tex]\lim_{\theta\rightarrow0}\frac{\sin{3\theta}}{\theta} = 3[/tex]

This is provable quite easily, I don't think I need to do that atm...

well, we know that

[tex]\lim_{\theta \rightarrow 0} \frac{\sin{\theta}}{\theta} = 1[/tex]

So that would imply that

[tex]3\lim_{\theta \rightarrow 0} \frac{\sin{\theta}}{\theta} = 3[/tex]

thus implying that

[tex]\lim_{\theta \rightarrow 0} \frac{3\sin{\theta}}{\theta} = 3[/tex]

By substitution

[tex]\lim_{\theta \rightarrow 0} \frac{\sin{3\theta}}{\theta} = \lim_{\theta \rightarrow 0} \frac{3\sin{\theta}}{\theta}[/tex]

(iffy step)

If you drop the limit on both sides...

[tex]\frac{\sin{3\theta}}{\theta}=\frac{3\sin{\theta}}{\theta}[/tex]

Which oh so quickly becomes

[tex]\sin{3\theta}=3\sin{\theta}[/tex]